Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection

TL;DR

This paper develops a method for nonlinear approximation and estimation using dyadic piecewise polynomials, effectively adapting to inhomogeneous and anisotropic smoothness in multivariate density estimation with linear computational complexity.

Contribution

It introduces a novel approximation and estimation framework that adapts to complex smoothness structures, extending to inhomogeneous and anisotropic classes including Besov spaces.

Findings

Achieves minimax adaptive estimation for complex smoothness classes.

Provides a computationally efficient estimation procedure.

Demonstrates theoretical optimality in approximation rates.

Abstract

This article is devoted to nonlinear approximation and estimation via piecewise polynomials built on partitions into dyadic rectangles. The approximation rate is studied over possibly inhomogeneous and anisotropic smoothness classes that contain Besov classes. Highlighting the interest of such a result in statistics, adaptation in the minimax sense to both inhomogeneity and anisotropy of a related multivariate density estimator is proved. Besides, that estimation procedure can be implemented with a computational complexity simply linear in the sample size.